 JEE Mains Previous Years Questions with Solutions

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1

AIEEE 2004

A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement $x$ is proportional to
A
$x$
B
${e^x}$
C
${x^2}$
D
${\log _e}x$

Explanation

Given that, retardation $\propto$ displacement

$\Rightarrow$ $a=-kx$

But we know $a = v{{dv} \over {dx}}\,\,\,\,\,\,\,\,\,$

$\therefore$ ${{vdv} \over {dx}} = - kx$

$\Rightarrow \int\limits_{{v_1}}^{{v_2}} v \,dv = - k\int\limits_0^x {xdx}$

$\left( {v_2^2 - v_1^2} \right) = - k{{{x^2}} \over 2}$

$\Rightarrow {1 \over 2}m\left( {v_2^2 - v_1^2} \right) = {1 \over 2}mk\left( {{{ - x^2} \over 2}} \right)$

$\therefore$ Loss in kinetic energy is proportional to ${x^2}$.

$\therefore$ $\Delta K \propto {x^2}$
2

AIEEE 2003

A body is moved along a straight line by a machine delivering a constant power. The distance moved by the body in time $'t'$ is proportional to
A
${t^{3/4}}$
B
${t^{3/2}}$
C
${t^{1/4}}$
D
${t^{1/2}}$

Explanation

We know that $F \times v =$ Power

According to the question, power is constant.

$\therefore$ $F \times v = c\,\,\,\,$ where $c=$ constant

$\therefore$ $m{{dv} \over {dt}} \times v = c$ $\,\,\,\,\left( \, \right.$ $\therefore$ $\left. {F = ma = {{mdv} \over {dt}}\,\,} \right)$

$\therefore$ $m\int\limits_0^v {vdv = c\int\limits_0^t {dt} } \,\,\,\,\,\,\,\,\,$ $\therefore$ ${1 \over 2}m{v^2} = ct$

$\therefore$ $v = \sqrt {{{2c} \over m}} \times {t^{{\raise0.5ex\hbox{1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{2}}}}$

${{dx} \over {dt}} = \sqrt {{{2c} \over m}} \times {t^{{\raise0.5ex\hbox{1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{2}}}}\,\,\,\,$ where $v = {{dx} \over {dt}}$

$\therefore$ $\int\limits_0^x {dx = \sqrt {{{2c} \over m}} } \times \int\limits_0^t {{t^{{\raise0.5ex\hbox{1} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{2}}}}} dt$
$x = \sqrt {{{2c} \over m}} \times {{2{t^{{\raise0.5ex\hbox{3} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{2}}}}} \over 3} \Rightarrow x \propto {t^{{\raise0.5ex\hbox{3} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{2}}}}$
3

AIEEE 2003

A spring of spring constant $5 \times {10^3}\,N/m$ is stretched initially by $5$ $cm$ from the unstretched position. Then the work required to stretch it further by another $5$ $cm$ is
A
$12.50$ $N$-$m$
B
$18.75$ $N$-$m$
C
$25.00$ $N$-$m$
D
$625$ $N$-$m$

Explanation

Given $k = 5 \times {10^3}N/m$

Work done when a spring stretched from x1 cm to x2 cm,

$W = {1 \over 2}k\left( {x_2^2 - x_1^2} \right)$

$= {1 \over 2} \times 5 \times {10^3}\left[ {{{\left( {0.1} \right)}^2} - {{\left( {0.05} \right)}^2}} \right]$

$= {{5000} \over 2} \times 0.15 \times 0.05 = 18.75\,\,Nm$
4

AIEEE 2003

A wire suspended vertically from one of its ends is stretched by attaching a weight of $200N$ to the lower end. The weight stretches the wire by $1$ $mm.$ Then the elastic energy stored in the wire is
A
$0.2$ $J$
B
$10$ $J$
C
$20$ $J$
D
$0.1$ $J$

Explanation

The elastic potential energy

$= {1 \over 2} \times$ Force $\times$ extension

$= {1 \over 2} \times 200 \times 0.001 = 0.1\,J$